I've got three, which I've arranged in increasing order of broad-reaching real-world significance.
One is pedagogical, by which I mean it doesn't really have much to do with trigonometry itself but is rather an artifact of the order in which mathematics tends to be taught. This is a motivation for the rigor of calculus. After all, up to this point in a person's education, the only functions $\mathbb{R}\to\mathbb{R}$ that they are likely to have discussed as such are polynomials, exponentials, logarithms, and trigonometric functions. It is possible using purely classical means to solve derivatives and integrals of low-degree polynomials. Therefore, trigonometric functions are a key insight into understanding the richness of function space, and therefore a motivation for wanting to investegate it in more than an ad-hoc way.
The second is more exciting, which is that trigonometry gives us a basic insight into the nature of the complex numbers. You may know that it is reasonable to append to the reals a special number $i$ which has the property that $i^2=-1$. The number is not a real number, but it turns out that all of the arithmetic operations that are possible on reals are also possible on numbers of the form $a+bi$. But notice what that form looks like. Just as the properties of real numbers invite us to plot them as points on a line, this understanding of the complex numbers invites us to plot them as points in the plane. Once we do this, a new thought (eventually) occurs to us: these complex numbers are vectors. But vectors have magnitude and direction. If one works through the technical difficulties, one finds* the beautiful relationship
$$\text{Mag}(z)e^{i\text{Ang}(z)} = \text{Mag}(z)\sin(\text{Ang}(z))+i\text{Mag}(z)\cos(\text{Ang}(z)).$$
The right hand side of that equation is of the form $a+bi$, which means we have discovered the relationship between the so called "polar" form of complex numbers and the way in which they are traditionally defined. But you might ask why we should care about complex numbers, well,
The third is more sobering, and it is what I would argue is the most widely manipulated phenomenon that higher mathematics is interested in (and in fact still maintains an interest in!): the concept of a Fourier series. The story here is rich and complex, and I do not know all of it and if you would like to learn, the rest of the internet has much better words to say than I do. But I will state the basic idea: any sufficiently nice function can be arbitrarily well-approximated by a function of the form:
$$f(x)\approx \hat f(x)=\sum_{i=0}^N a_n\cos(nx)+b_n\sin(nx)$$
This is perhaps surprising, but if you fiddle with some explicit approximations then you could become convinced of this eventually. What this means physically is that a properly encoded device could read the various $a_n$ and $b_n$ and reproduce the function arbitrarily well. More to the point, it means that if you want to tell your function to someone else (which in general contains an infinite amount of information), they can get arbitrarily fine detail if you just send them a finite number of the coefficients. Therefore, one can send signals over long distances with relative ease, and this is the method that makes radio, telephone, and the internet work. So the reality is that little trigonometry, with humble origins on the unit circle, is now responsible for making our "information society" possible.
(Ah, I did promise a connection to complex numbers. But perhaps you see it already: just as sines/cosines can be written as complex numbers, the opposite is true as well. In fact, this ends up being the "right" way to think of Fourier series mathematically, in that it admits incredibly, almost unbelievably, far-reaching generalizations. In the interest of keeping your attention, I have omitted these; they are quite technical and to fill you in on the background details would have more than tripled the size of this answer.)
(*In fact the story is slightly more complicated than this: the truth is that this is how one traditionally defines complex exponentials, but there are good reasons for doing so: things which one wants to be true about exponentials are true for this particular combination of sines and cosines, except for some technical obstructions that in any case no reasonable definition would be able to overcome.)
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Tomography, which is now used heavily nowadays as a diagnostic tool, relies on rather deep mathematics to work properly. There is in particular the Radon transform and its inverse, which are useful for reconstructing a three-dimensional visualization of body parts from "slices" taken by a CAT scanner.